Word problem related to ratio and proportion

Question

Problem:- Rs $3810$ are to be distributed among A,B,C and D, so that half of A's share is equal to one third of B's share. Also one forth of C's share is equal to one sixth of D's share.Find A's share.

Solution:- $\frac{A}{2} =\frac{B}{3} $

or $\frac{A}{B} =\frac{2}{3} $

Also $\frac{C}{4} =\frac{D}{6} $

or $\frac{C}{D} =\frac{2}{3} $

$A+B+C+D=3810$

$2x+3x+2y+3y=3810$

$x+y=762$

What to do next ??

please help

Answer 1

Something's missing here.

You talk about $A$'s part related to $B$'s. You talk also about $C$'s part related to $D$'s. But you can slice up which chunk goes to $A,B$ however you like, and give the rest to $C,D$, and satisfy the equations.

So there's any number of solutions as to what $A$'s share is.

Answer 2

The problem is this system is under-determined so we need to treat one variable as known. If you do this (for A assumed to be known) you can obtain a family of solutions as;

$ B = 3A/2 \quad C=1524 - A \quad D=2286-3A/2 $

Then we pick $A$ appropriately (non-negative, less than total sum, etc).

Answer 3

HINT:

Letting

$$A,B,C,D \quad \text{be} \quad 2p,3p,4q,6q,$$

we have to solve

$$ 5p+10q = 3810,\quad \text{or,} \quad p/2 + q = 381 $$

as a Diophantine equation as one option.